Is the Given Map Continuous and Bijective for Cantor Sets?

[ljw5021]

Homework Statement

Consider the map phi : C -> I which maps each point of the middle third Cantor set C, considered as a subset of real numbers between 0 and 1 written in base 3 and containing only digits 0 and 2, to the set of real numbers I=[0,1] written in base 2, according to the rule: 0.a_1a_2a_3... -> 0.b_1b_2b_3... where b_i = a_i / 2

(1) Prove that phi is a continuous map of C onto I.
(2) Prove that phi is not bijective.

Homework Equations

The Attempt at a Solution

I decided to use the definition of continuity that all open sets in I must have open pre-images in C. I tried saying pick (1/2,1) in I (that is, all elements of the form 0.1b_2b_3...). This open intervals pre-image in C would be the intersection of (2/3,1) with the cantor set, C.

It's really sketchy in my head and I would love some help. Also proving that phi is not bijective. I feel it may have something to do with certain decimal expansions.

Thanks.

 

[Billy Bob]

There are more elegant ways to do this, but I would just get my hands dirty and prove phi is continuous at each x in the domain, the boring epsilon-delta way.

Here is a hint, by looking at how to prove phi is continuous at x=1/4. Note 1/4 has ternary expansion 0.020202... , so phi(1/4)=0.010101... in binary, i.e. 1/3.

Let epsilon > 0.

Find N such that 1/2^N < epsilon. For this hint, suppose N=5 works.

(Note: I'm not going to be careful and figure out whether I meant N or N-1 or N+1 or N+2 in this hint.)

So go out to the fifth (or sixth) binary "decimal" place in f(1/4)=1/3, so you have 0.010101..., and furthermore L < 1/3 < U, where L=0.01010100000000... and U= 0.01010111111111... (binary).

Note U-L<1/2^5 (or 6 or whatever).

Now L and U are the images of 0.020202000000... and 0.0202022222222... (ternary) respectively, and these two numbers differ by 0.00000022222... (ternary) which is 1/3^5 (or 6 or 7), so now you have your delta (or divide it by 2).

Just write this up in general.

You may or may not have to make a special case if x ends with 0 repeating or 2 repeating (ternary). For example 1/3 = 0.100000... = 0.0222222... (ternary) so 1/3 is in C. What is phi(1/3)? Also 2/3 = 0.12222222... = 0.2000000... (ternary) so 2/3 is in C. What is phi(2/3)?